Calculate shipment cost without losing units at transfer points

Agree what one unit and one route mean

Google’s minimum-cost flow formulation combines per-unit arc costs, capacity limits and node balances. Use outgoing minus incoming as net supply throughout this worksheet.

The fictional object is one standardized display kit. S is its source, H a relay, and T the destination. Every amount refers to the same kit; there is no repacking, damage allowance or partial delivery. All costs are teaching currency units per kit per leg. These inputs are invented, not a carrier tariff.

For an actual proposal, ask the logistics owner to verify the physical meaning of each route. A line on a diagram does not establish that a carrier accepts the item, that a transfer point can handle it or that an arrival promise is supported. Retain those questions with the diagram rather than hiding them inside an unexplained cost adjustment.

Reconcile every node and every leg

NetworkX documents net inflow as demand; that convention is opposite to our net-supply convention. Five kits leave S and are required at T. Arcs S–H and H–T each have capacity 3 and cost 1; S–T has capacity 5 and cost 4. Send 3 via H and 2 directly. S balance: 3 + 2 − 0 = 5. H: 3 − 3 = 0. T: 0 − (3 + 2) = −5. Leg costs are 3 × 1 = 3, 3 × 1 = 3 and 2 × 4 = 8; total 14.

Put the two relay legs on separate lines, even when they happen to have the same quantities and charges. Ask a reviewer to trace a physical kit through the diagram and identify where each charge belongs. The explanation should make sense without relying on a software field name.

Keep a legend next to the record when handing it to another team. The numerical result alone is insufficient if the receiving analyst interprets the signs differently. Do not repair a disagreement by changing the destination requirement; resolve the representation and retain the original inputs.

Explain the capacity-driven alternative

For relay quantities 0, 1, 2 and 3, direct quantities are 5, 4, 3 and 2; total costs are 20, 18, 16 and 14. If H–T capacity becomes 2, the last row is unavailable. The best remaining cost is 16. New balances are S: 2 + 3 = 5; H: 2 − 2 = 0; T: −(2 + 3) = −5.

Present this as a separate operating scenario, not an unexplained replacement for the earlier answer. Ask what evidence supports the revised restriction and whether it applies during the relevant movement window. An outdated limit and a newly confirmed limit should not be treated as equally reliable facts.

The example deliberately excludes fixed booking charges, commercial discounts and timing commitments. If an actual quote contains them, ask the owner for a model that represents those terms. Do not spread an inconvenient fixed charge across an arbitrary quantity merely to preserve the teaching calculation. A neat number is not more useful than an honest statement that the commercial question has changed.

Keep the recommendation distinct from execution

Attach the diagram, owner-supplied route evidence and cost interpretation to the proposed plan. The reviewer should be able to challenge the physical route independently of the arithmetic. A mathematically coherent proposal may still be unsuitable because of handling restrictions or a customer commitment absent from the exercise.

Identify the unresolved fact that would most change the recommendation. Request its verification through the authorized logistics owner rather than contacting carriers, booking transport or promising a delivery. Keep any operational authorization outside this teaching record.

If the actual arrangement later differs, preserve the previous proposal before recomputing. Separate a changed commercial term from a clerical mistake or a newly unavailable route. That distinction helps a team learn why the recommendation moved without pretending that the original calculation anticipated information it did not contain.

Make the reasoning concrete.

A retail display coordinator receives a cheaper-looking transfer proposal without an explanation of the relay.

The coordinator prepares the record for review and makes no shipment or payment.

Take the questions into your next brief.

Use these prompts in your own notes. Nothing is submitted from this page.

1. Object

   What physical item does one unit represent?

2. Route

   Who verifies that each leg is operationally available?

3. Meaning

   Can the recipient interpret every field without guessing?

4. Terms

   Which commercial condition is absent from the illustration?

5. Review

   What evidence could overturn the recommendation?

Watch for the wrong conclusion.

• Mixing kits and cartons without an explicit conversion.
• Treating a drawn route as proof of carrier acceptance.
• Hiding a fixed commercial charge inside an invented average.

Sources & scope

The decision reasoning and fictional inputs are original. Method statements and mathematical applications remain attributed, including later restatements. Documentation was read on the stated date; no installed solver version is asserted. Local exhaustive arithmetic was executed, not real operations. No purchasing, staffing, shipment, scheduling or account change is authorized.

• Minimum Cost Flows →Primary documentation · Supports: Supports the stated model.
• min_cost_flow →Primary documentation · Supports: Supports the mathematical formulation.